New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method

Discrete fractional calculus ([Formula: see text]) is significant for neural networks, complex dynamic systems and frequency response analysis approaches. In contrast with the continuous-time frameworks, fewer outcomes are accessible for discrete fractional operators. This study investigates some major consequences of two sorts of inequalities by considering discrete Atangana–Baleanu [Formula: see text]-fractional operator having [Formula: see text]-discrete generalized Mittag-Leffler kernels in the sense of Riemann type ([Formula: see text]). Certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete [Formula: see text]-fractional operators having [Formula: see text]-discrete generalized Mittag-Leffler kernels are given. Moreover, several other generalizations can be generated for nabla [Formula: see text]-fractional sums. The proposing discretization is a novel form of the existing operators that can be provoked by some intriguing features of chaotic systems to design efficient dynamics description in short time domains. Furthermore, by combining two mechanisms, numerous new special cases are introduced. 

SOME RECENT DEVELOPMENTS ON DYNAMICAL ℏ-DISCRETE FRACTIONAL TYPE INEQUALITIES IN THE FRAME OF NONSINGULAR AND NONLOCAL KERNELS

Solitons, Nonlinear Evolution Equations and Inverse Scattering: Remarks on Riemann-Hilbert problems

In the current paper, we carry out an investigation into the exact solutions of the (3+1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov (fractional mKdV–ZK) equation. Based on the conformable fractional derivative and its properties, the fractional mKdV–ZK equation is reduced into an ordinary differential equation which has been solved analytically by the variable separated ODE method. Various types of analytic solutions in terms of hyperbolic functions, trigonometric functions and Jacobi elliptic functions are derived. All conditions for the validity of all obtained solutions are given

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Solitons and other solutions of (3 + 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation

Finding the exact solutions of nonlinear fractional differential equations has gained considerable attention, during the past two decades. In this paper, the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities are studied. Several exact soliton solutions, including the bright (non-topological) and singular soliton solutions are formally extracted by making use of the ansatz method. Results demonstrate that the method can efficiently handle the time-fractional Klein–Gordon equations with different nonlinearities.

Bright and singular soliton solutions of the conformable time-fractional Klein–Gordon equations with different nonlinearities

On the conformable nonlinear schrödinger equation with second order spatiotemporal and group velocity dispersion coefficients

In this study, the nonlinear fractional partial differential equations have been defined by the modified Riemann–Liouville fractional derivative. By using this fractional derivative and traveling wave transformation, the nonlinear fractional partial differential equations have been converted into nonlinear ordinary differential equations. The modified trial equation method is implemented to obtain exact solutions of the nonlinear fractional Klein–Gordon equation and fractional clannish random walker's parabolic equation. As a result, some exact solutions including single kink solution and periodic and rational function solutions of these equations have been successfully obtained. Copyright © 2015 John Wiley & Sons, Ltd.

Meryem Odabasi

Papers

On the conformable nonlinear schrödinger equation with second order spatiotemporal and group velocity dispersion coefficients

On the solutions of the nonlinear fractional differential equations via the modified trial equation method

Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations

Analytical solutions of some nonlinear fractional-order differential equations by different methods

A note on the traveling wave solutions of some nonlinear evolution equations